Fanography

A tool to visually study the geography of Fano 3-folds.

Identification

Fano variety 3-8

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup

rank
3 (others)
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.


This variety is the blowup of

  • 2-24, in a curve of genus 0
  • 2-34, in a curve of genus 0
Deformation theory
number of moduli
3

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}$ 1 0
$0$ 0 3